conjugation (mnemonic)

In pre-college mathematics, students typically learn how to rationalize the denominator (or, in some cases, numerator) of expressions such as ${\displaystyle \frac{3}{\sqrt{11}+2}}$ and ${\displaystyle \frac{\sqrt{x+h}-\sqrt{x-h}}{2h}}$. In to do this, they multiply the numerator and denominator of the fraction by an algebraic conjugate (or, in some cases, its negative) to eliminate the square root(s) (http://planetmath.org/SquareRoot) in the appropriate part of the fraction. Typically, the only algebraic conjugates that pre-college students encounter are those in some quadratic extension.

Most students who have advanced far enough in mathematics to encounter rationalizing denominators or numerators have also encountered some (usually Indo-European) foreign . Such students are familiar with the concept of of verbs, in which the ending of the verb changes to make agreement with the person and number of the subject. A helpful mnemonic for students to the algebraic conjugates that they need to use is pointing out to them that the procedure in mathematics is (and actually easier) than in foreign . The algebraic conjugates (or their negatives) that they need are nothing more than changing the ending of the number. For example, the way that a pre-college student is taught to rationalize the denominator of an expression such as ${\displaystyle \frac{3}{\sqrt{11}+2}}$ is: